SpringerLink
Log in

An Improved Trudinger–Moser Inequality Involving N-Finsler–Laplacian and Lp Norm

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

Let \(F: \mathbb {R}^{N} \rightarrow [0, +\infty )\) be a convex function of class \(C^{2}(\mathbb {R}^{N} \backslash \{0\})\), which is even and positively homogeneous of degree 1. Let \({\Omega }\subset \mathbb {R}^{N}(N\geq 2)\) be a smooth bounded domain, we denote \(\gamma _{1}=\inf \limits _{u\in W^{1, N}_{0}({\Omega })\backslash \{0\}}\frac {{\int \limits }_{\Omega }F^{N}(\nabla u)dx}{\| u\|_{p}^{N}}\) and define \(\|u\|_{N,F,\gamma , p}=\left ({\int \limits }_{\Omega }F^{N}(\nabla u)dx-\gamma \| u\|_{p}^{N}\right )^{\frac {1}{N}}.\) Then for p > 1 and 0 ≤ γ < γ1, we have

$$ \sup_{u\in W^{1, N}_{0}({\Omega}), \|u\|_{N,F,\gamma, p}\leq 1}{\int}_{\Omega}e^{\lambda_{N} |u|^{\frac{N}{N-1}}}dx<+\infty, $$

where \(\lambda _{N}=N^{\frac {N}{N-1}} \kappa _{N}^{\frac {1}{N-1}}\) and κN is the volume of a unit Wulff ball in \(\mathbb {R}^{N}\). Moreover, by using blow-up analysis and capacity technique, we prove that the supremum can be attained for any 0 ≤ γ < γ1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

The data that supports the findings of this study are available within the article.

References

  1. Adachi, S., Tanaka, K.: Trudinger type inequalities in \(\mathbb {R}^{N}\) and their best exponents. Proc. Amer. Math. Soc. 128, 2051–2057 (2000)

    Article  MathSciNet  Google Scholar 

  2. Adams, D.: A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. 128, 385–398 (1988)

    Article  MathSciNet  Google Scholar 

  3. Adimurthi, A., Druet, O.: Blow-up analysis in dimension 2 and a sharp form of Moser–Trudinger inequality. Comm. Partial Differ. Equ. 29, 295–322 (2004)

    Article  Google Scholar 

  4. Adimurthi, A., Yang, Y.: An interpolation of Hardy inequality and Trudinger–Moser inequality. Int. Math. Res. Notices. 13, 2394–2426 (2010)

    Google Scholar 

  5. Alvino, A., Ferone, V., Trombetti, G., Lions, P.: Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 275–293 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  6. Belloni, M., Ferone, V., Kawohl, B.: Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators. Z. Angew. Math. Phys. 54, 771–783 (2003)

    Article  MathSciNet  Google Scholar 

  7. Cao, D.: Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb {R}^{2}\). Comm. Partial Differ. Equ. 17, 407–435 (1992)

    Article  MathSciNet  Google Scholar 

  8. Carleson, L., Chang, S.Y.A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 110, 113–127 (1986)

    MathSciNet  Google Scholar 

  9. Černý, R., Cianchi, A., Henel, S.: Concentration-compactness principles for Moser–Trudinger inequalities: new results and proofs. Ann. Mat. Pura Appl. 192, 225–243 (2013)

    Article  MathSciNet  Google Scholar 

  10. Chang, S.Y.A., Yang, P.: The inequality of Moser and Trudinger and applications to conformal geometry. Comm. Pure Appl. Math. 56, 1135–1150 (2003)

    Article  MathSciNet  Google Scholar 

  11. Chen, L., Lu, G., Zhu, M.: Existence and nonexistence of extremals for critical Adams inequalities in \(\mathbb {R}^{4}\) and Trudinger–Moser inequalities in \(\mathbb {R}^{2}\). Adv. Math. 368, 107143 (2020)

    Article  MathSciNet  Google Scholar 

  12. Chen, L., Lu, G., Zhu, M.: Sharpened Trudinger–Moser inequalities on the Euclidean space and Heisenberg group. J. Geom. Anal. 31(12), 12155–12181 (2021)

    Article  MathSciNet  Google Scholar 

  13. Chen, L., Lu, G., Zhu M.: A sharpened form of Adams type inequalities on higher order Sobolev spaces \(W^{m,\frac {n}{m}}(\mathbb {R}^{n})\): a simple approach. Canadian Mathematical Bulletin 65(4), 895–905 (2022). https://doi.org/10.4153/S0008439521001028

    Article  MathSciNet  Google Scholar 

  14. Csató, G., Roy, P.: Extremal functions for the singular Moser–Trudinger inequality in 2 dimensions. Calc. Var. Partial Differ. Equ. 54, 2341–2366 (2015)

    Article  MathSciNet  Google Scholar 

  15. Csató, G., Roy, P., Nguyen, V.H.: Extremals for the singular Moser–Trudinger inequality via n-harmonic transplantation. J. Differ. Equ. 270, 843–882 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  16. de Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in \(\mathbb {R}^{2}\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3, 139–153 (1995)

    Article  MathSciNet  Google Scholar 

  17. de Souza, M., do, Ó.J.M.: A sharp Trudinger–Moser type inequality in \(\mathbb {R}^{2}\). Trans. Amer. Math. Soc. 366, 4513–4549 (2014)

    Article  MathSciNet  Google Scholar 

  18. do, Ó.J.M.: N-Laplacian equations in \(\mathbb {R}^{N}\) with critical growth. Abstr. Appl. Anal. 2, 301–315 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  19. do, Ó.J.M., de Souza, M., de Medeiros, E., Severo, U.: An improvement for the Trudinger–Moser inequality and applications. J. Differ. Equ. 256, 1317–1349 (2014)

    Article  MathSciNet  Google Scholar 

  20. do, Ó.J.M., de Souza, M.: A sharp inequality of Trudinger–Moser type and extremal functions in \(h^{1,n}(\mathbb {R}^{n})\). J. Differ. Equ. 258, 4062–4101 (2015)

    Article  ADS  Google Scholar 

  21. Ferone, V., Kawohl, B.: Remarks on a Finsler–Laplacian. Proc. Amer. Math. Soc. 137, 247–253 (2009)

    Article  MathSciNet  Google Scholar 

  22. Flucher, M.: Extremal functions for Trudinger–Moser inequality in 2 dimensions. Comment. Math. Helv. 67, 471–497 (1992)

    Article  MathSciNet  Google Scholar 

  23. Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119, 125–136 (1991)

    Article  MathSciNet  Google Scholar 

  24. Heinonen, J., Kilpelainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford, University Press Oxford (1993)

    Google Scholar 

  25. Lam, N., Lu, G.: Sharp Adams type inequalities in Sobolev spaces \(W^{m,\frac {n}{m}}(\mathbb {R}^{n})\) for arbitrary integer m. J. Differ. Equ. 253(4), 1143–1171 (2012)

    Article  ADS  Google Scholar 

  26. Lam, N., Lu, G.: A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument. J. Differ. Equ. 255, 298–325 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  27. Lam, N., Lu, G.: Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition. J. Geom. Anal. 24, 118–143 (2014)

    Article  MathSciNet  Google Scholar 

  28. Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities. Rev. Mat. Iberoam. 33, 1219–1246 (2017)

    Article  MathSciNet  Google Scholar 

  29. Lam, N., Lu, G., Zhang, L.: Sharp singular Trudinger–Moser inequalities under different norms. Adv. Nonlinear Stud. 19(2), 239–261 (2019)

    Article  MathSciNet  Google Scholar 

  30. Lam, N., Lu, G., Zhang, L.: Existence and nonexistence of extremal functions for sharp Trudinger–Moser inequalities. Adv. Math. 352, 1253–1298 (2019)

    Article  MathSciNet  Google Scholar 

  31. Li, J., Lu, G., Zhu, M.: Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions. Calc. Var. Partial Differential Equations 57(3), Article ID 84 (2018)

  32. Li, J., Lu, G., Zhu, M.: Concentration-compactness principle for Trudinger–Moser’s inequalities on Riemannian manifolds and Heisenberg groups: a completely symmetrization-free argument. Adv. Nonlinear Stud. 21(4), 917–937 (2021)

    Article  MathSciNet  Google Scholar 

  33. Li, X., Yang, Y.: Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space. J. Differ. Equ. 264, 4901–4943 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  34. Li, Y.: Moser–trudinger inequality on compact Riemannian manifolds of dimension two. J. Partial Differ. Equ. 14, 163–192 (2001)

    MathSciNet  Google Scholar 

  35. Li, Y.: Extremal functions for the Moser–Trudinger inequalities on compact Riemannian manifolds. Sci. China Ser. A 48(5), 618–648 (2005)

    Article  MathSciNet  Google Scholar 

  36. Li, Y.: Remarks on the extremal functions for the Moser–Trudinger inequality. Acta. Math. Sin. (Engl. Ser.) 22(2), 545–550 (2006)

    Article  MathSciNet  Google Scholar 

  37. Li, Y., Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \(\mathbb {R}^{N}\). Indiana Univ. Math. J. 57, 451–480 (2008)

    Article  MathSciNet  Google Scholar 

  38. Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)

    Article  MathSciNet  Google Scholar 

  39. Lin, K.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc. 348, 2663–2671 (1996)

    Article  MathSciNet  Google Scholar 

  40. Lions, P.L.: The concentration-compactness principle in the calculus of variations. Part I Rev. Mat. Iberoamericana 1, 145–201 (1985)

    Article  MathSciNet  Google Scholar 

  41. Lu, G., Tang, H.: Sharp singular Trudinger–Moser inequalities in Lorentz-Sobolev spaces. Adv. Nonlinear Stud. 16, 581–601 (2016)

    Article  MathSciNet  Google Scholar 

  42. Lu, G., Tang, H.: Sharp Moser–Trudinger inequalities on hyperbolic spaces with exact growth condition. J. Geom. Anal. 26(2), 837–857 (2016)

    Article  MathSciNet  Google Scholar 

  43. Lu, G., Zhu, M.: A sharp Trudinger–Moser type inequality involving Ln norm in the entire space \(\mathbb {R}^{n}\). J. Differ. Equ. 267(5), 3046–3082 (2019)

    Article  ADS  Google Scholar 

  44. Lu, G., Yang, Y.: The sharp constant and extremal functions for Moser-Trudinger inequalities involving lp norms. Discrete Contin. Dyn. Syst. 25, 963–979 (2009)

    Article  MathSciNet  Google Scholar 

  45. Malchiodi, A., Martinazzi, L.: Critical points of the Moser–Trudinger functional on a disk. J. Eur. Math. Soc. 16, 893–908 (2014)

    Article  MathSciNet  Google Scholar 

  46. Mancini, G., Martinazzi, L.: The Moser–Trudinger inequality and its extremals on a disk via energy estimates. Calc. Var. Partial Differ. Equ. 20, 56–94 (2017)

    MathSciNet  Google Scholar 

  47. Masmoudi, N., Sani, F.: Trudinger–Moser inequalities with the exact growth condition in \(\mathbb {R}^{n}\) and application. Commun. Partial Differ. Equ. 40, 1408–1440 (2015)

    Article  MathSciNet  Google Scholar 

  48. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970)

    Article  MathSciNet  Google Scholar 

  49. Nguyen, V.H.: Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions. Ann. Global Anal. Geom. 54, 237–256 (2018)

    Article  MathSciNet  Google Scholar 

  50. Peetre, J.: Espaces d’interpolation et theoreme de Soboleff. Ann. Inst. Fourier (Grenoble) 16, 279–317 (1966)

    Article  MathSciNet  Google Scholar 

  51. Pohožaev, S.: The Sobolev embedding in the special case pl = n. In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965, Mathematics Sections, Moscov. Energet. Inst. Moscow, pp. 158–170 (1965)

  52. Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \(\mathbb {R}^{2}\). J. Funct. Anal. 219, 340–367 (2005)

    Article  MathSciNet  Google Scholar 

  53. Ruf, B., Sani, F.: Sharp Adams-type inequalities in \(\mathbb {R}^{n}\). Trans. Amer. Math. Soc. 365, 645–670 (2013)

    Article  MathSciNet  Google Scholar 

  54. Struwe, M.: Critical points of embeddings of \(h_{0}^{1, n}\) into Orlicz spaces. Ann. Inst. H. Poincaré, Anal. Non Linéaire 5, 425–464 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  55. Struwe, M.: Positive solution of critical semilinear elliptic equations on non-contractible planar domain. J. Eur. Math. Soc. 2, 329–388 (2000)

    Article  MathSciNet  Google Scholar 

  56. Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3, 697–718 (1976)

    MathSciNet  Google Scholar 

  57. Tintarev, C.: Trudinger–moser inequality with remainder terms. J. Funct. Anal. 266, 55–66 (2014)

    Article  MathSciNet  Google Scholar 

  58. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)

    MathSciNet  Google Scholar 

  59. Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  60. Wang, G., **a, C.: A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch. Ration. Mech. Anal. 99, 99–115 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  61. Wang, G., **a, C.: Blow-up analysis of a Finsler–Liouville equation in two dimensions. J. Differ. Equ. 252, 1668–1700 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  62. **e, R., Gong, H.: A priori estimates and blow-up behavior for solutions of − QNu = V eu in bounded domain in \(\mathbb {R}^{N}\). Sci. China Math. 59, 479–492 (2016)

    Article  MathSciNet  Google Scholar 

  63. Yang, Y.: A sharp form of Moser–Trudinger inequality in high dimension. J. Funct. Anal. 239, 100–126 (2006)

    Article  MathSciNet  Google Scholar 

  64. Yang, Y.: Corrigendum to: A sharp form of Moser–Trudinger inequality in high dimension. J. Funct. Anal. 242, 669–671 (2007)

    Article  MathSciNet  Google Scholar 

  65. Yang, Y.: A sharp form of the Moser–Trudinger inequality on a compact Riemannian surface. Trans. Amer. Math. Soc. 359, 5761–5776 (2007)

    Article  MathSciNet  Google Scholar 

  66. Yang, Y.: Extremal functions for Trudinger–Moser inequalities of Adimurthi-Druet type in dimension two. J. Differ. Equ. 258, 3161–3193 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  67. Yang, Y., Zhu, X.: Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two. J. Funct. Anal. 272, 3347–3374 (2017)

    Article  MathSciNet  Google Scholar 

  68. Yudovič, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations, (Russian). Dokl. Akad. Nauk SSSR 138, 805–808 (1961)

    MathSciNet  Google Scholar 

  69. Zhou, C.L., Zhou, C.Q.: Moser–Trudinger inequality involving the anisotropic Dirichlet norm \(({\int \limits }_{{{\Omega }}}f^{N}(\nabla u)dx)^{\frac {1}{N}}\) on \(w_{0}^{1, N}({{\Omega }})\). J. Funct. Anal. 276, 2901–2935 (2019)

    Article  MathSciNet  Google Scholar 

  70. Zhou, C.L.: Anisotropic Moser–Trudinger inequality involving Ln norm. J. Differential Equations 268, 7251–7285 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  71. Zhu, J.: Improved Moser–Trudinger inequality involving Lp norm in n dimensions. Adv. Nonlinear Stud. 14, 273–293 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (12201089), the Natural Science Foundation Project of Chongqing (CSTB2022NSCQ-MSX0226), the Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202200513) and Chongqing Normal University Foundation (21XLB039).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Yanjun Liu

Authors
  1. Yanjun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanjun Liu.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Y. An Improved Trudinger–Moser Inequality Involving N-Finsler–Laplacian and Lp Norm. Potential Anal 60, 673–701 (2024). https://doi.org/10.1007/s11118-023-10066-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-023-10066-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation